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Properties

Label 2-450-225.92-c1-0-10
Degree $2$
Conductor $450$
Sign $0.886 + 0.462i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.838 − 0.544i)2-s + (−1.49 − 0.869i)3-s + (0.406 − 0.913i)4-s + (0.466 + 2.18i)5-s + (−1.72 + 0.0870i)6-s + (3.77 − 1.01i)7-s + (−0.156 − 0.987i)8-s + (1.48 + 2.60i)9-s + (1.58 + 1.57i)10-s + (−1.19 + 5.61i)11-s + (−1.40 + 1.01i)12-s + (1.67 − 2.57i)13-s + (2.61 − 2.90i)14-s + (1.20 − 3.68i)15-s + (−0.669 − 0.743i)16-s + (−1.48 + 0.235i)17-s + ⋯
L(s)  = 1  + (0.593 − 0.385i)2-s + (−0.864 − 0.501i)3-s + (0.203 − 0.456i)4-s + (0.208 + 0.977i)5-s + (−0.706 + 0.0355i)6-s + (1.42 − 0.382i)7-s + (−0.0553 − 0.349i)8-s + (0.496 + 0.868i)9-s + (0.500 + 0.499i)10-s + (−0.360 + 1.69i)11-s + (−0.405 + 0.293i)12-s + (0.464 − 0.715i)13-s + (0.698 − 0.776i)14-s + (0.310 − 0.950i)15-s + (−0.167 − 0.185i)16-s + (−0.361 + 0.0571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.886 + 0.462i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72579 - 0.423138i\)
\(L(\frac12)\) \(\approx\) \(1.72579 - 0.423138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.838 + 0.544i)T \)
3 \( 1 + (1.49 + 0.869i)T \)
5 \( 1 + (-0.466 - 2.18i)T \)
good7 \( 1 + (-3.77 + 1.01i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.19 - 5.61i)T + (-10.0 - 4.47i)T^{2} \)
13 \( 1 + (-1.67 + 2.57i)T + (-5.28 - 11.8i)T^{2} \)
17 \( 1 + (1.48 - 0.235i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-2.64 + 3.63i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-8.64 + 0.452i)T + (22.8 - 2.40i)T^{2} \)
29 \( 1 + (0.823 + 7.83i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.0536 + 0.510i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (3.48 - 1.77i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-0.721 - 3.39i)T + (-37.4 + 16.6i)T^{2} \)
43 \( 1 + (-2.36 - 8.83i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.57 - 2.08i)T + (9.77 + 45.9i)T^{2} \)
53 \( 1 + (11.1 + 1.76i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-0.683 + 0.145i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (11.1 + 2.36i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (0.986 - 0.798i)T + (13.9 - 65.5i)T^{2} \)
71 \( 1 + (3.51 + 4.83i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.73 - 2.41i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (9.91 - 1.04i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (2.49 + 0.958i)T + (61.6 + 55.5i)T^{2} \)
89 \( 1 + (-3.42 - 10.5i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.77 + 3.42i)T + (-20.1 - 94.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.10292511344420217421364952482, −10.59652685292283968512252698041, −9.624281811126236809593540041437, −7.80166970051487028176939304121, −7.26876549969579005219431586206, −6.30237026204440745445599342845, −5.07647623854924538374903248338, −4.52448202175463033391367544877, −2.67681166968618332013096785028, −1.48827891123115201545934990627, 1.35156042976466530551159457024, 3.53144341005552392029420840096, 4.77272747196805469974999841013, 5.33197504669239912254171337632, 6.02044664554176061451100319452, 7.39138849241007487696296293881, 8.722410734604014468785648714531, 8.920508375884872478161280373860, 10.68822891711757450978541418639, 11.24063352974407706747881038321

Graph of the $Z$-function along the critical line